The present disclosure generally relates to identification of mistuning in rotating, bladed structures, and, more particularly, to the development and use of reduced order models as an aid to the identification of mistuning.
It is noted at the outset that the term “bladed disk” is commonly used to refer to any (blade-containing) rotating or non-rotating part of an engine or rotating apparatus without necessarily restricting the term to refer to just a disk-shaped rotating part. Thus, a bladed disk can have externally-attached or integrally-formed blades or any other suitable rotating protrusions. Also, this rotating mechanism may have any suitable shape, whether in a disk form or not. Further, the term “bladed disk” may include stators or vanes, which are non-rotating bladed disks used in gas turbines. Various types of devices such as fans, pumps, turbochargers, compressors, engines, turbines, and the like, may be commonly referred to as “rotating apparatus.”
FIG. 1 illustrates a bladed disk 10 which is representative of those used in gas turbine engines. One such exemplary gas turbine 12 is illustrated in FIG. 1. Bladed disks used in turbine engines are nominally designed to be cyclically symmetric. If this were the case, then all blades would respond with the same amplitude when excited by a traveling wave. However, in practice, the resonant amplitudes of the blades are very sensitive to small changes in their properties. The small variations that result from the manufacturing process and wear cause some blades to have a significantly higher response and may cause them to fail from high cycle fatigue (HCF). This phenomenon is referred to as the mistuning problem, and has been studied extensively.
FIG. 2 represents an exemplary selection of nodal diameter modes 13-15 in a bladed disk. In the zero nodal diameter mode 13 (part (a) in FIG. 2), all the blades move in phase with one another, while in the higher nodal diameter modes 14-15, the blades move out of phase. FIG. 2(b) illustrates a mode 14 with five (5) nodal diameters, whereas FIG. 2(c) illustrates a mode 15 with ten (10) nodal diameters. The displacement of the blades as a function of angular position is given in these modes by functions sin(nθ) and cos(nθ), where “n” defines the number of nodal diameters. For a given value of “n”, the corresponding sine and cosine modes both have the same natural frequency. The only nodal diameter modes which do not have repeated frequencies are the cases of n=0 and n=N/2, where N is the number of blades on the disk.
FIG. 3 is an exemplary nodal diameter map 16 of a bladed disk's natural frequencies. Thus, the natural frequencies of a bladed disk are plotted as a function of the number of nodal diameters in their corresponding mode. When plotted in this fashion, the frequencies cluster into families of modes. Each family consists of a set of N nodal diameter modes. Each mode 17-19 at the right in FIG. 3 represents the blade deformation in the corresponding family. Although the relative amplitudes of the blades varies from one nodal diameter to the next, the deformation within a given blade remains uniform throughout all modes within a given family, at least for families which are isolated in frequency from their neighbors. The blade deformation in the first few families generally resembles the simple bending and torsion modes of a cantilevered plate. Many families of modes have most of their strain energy stored in the blades. These families appear relatively flat in FIG. 3, because the added strain energy introduced with additional nodal diameters has a minimal effect on their natural frequency. In contrast, mode families with large strain energy in the disk increase their frequency rapidly from one nodal diameter to the next.
Mistuning can significantly affect the vibratory response of a bladed disk. This sensitivity stems from the nature of the eigenvalue problem that describes a disk's modes and natural frequencies. An eigenvalue is equal to the square of the natural frequency of a mode. The eigenvalues of a bladed disk are inherently closely spaced due to the system's rotationally periodic design, as can be seen from the plot in FIG. 3. Therefore, the eigenvectors (mode shapes) of a bladed disk are very sensitive to the small perturbations caused by mistuning. In the case of very small mistuning, the blade displacements in the modes are given by distorted sine and cosine waves, while large mistuning can alter the modes to such an extent that the majority of the motion will be localized to just one or two blades. FIG. 4 illustrates exemplary forced response tracking plots 20-21 of a tuned bladed disk system (plot 20) and the mistuned disk system (plot 21). The plot 21 illustrates blade amplitude magnification caused by mistuning.
To address the mistuning problem, researchers have developed reduced order models (ROMs) of the bladed disk. These ROMs have the structural fidelity of a finite element model of the full rotor, while incurring computational costs that are comparable to that of a mass-spring model. In numerical simulations, most published ROMs have correlated extremely well with numerical benchmarks. However, some models have at times had difficulty correlating with experimental data. These results suggest that the source of the error may lie in the inability to determine the correct input parameters to the ROMs.
The standard method of measuring mistuning in rotors with attachable blades is to mount each blade in a broach block and measure its natural frequency. The difference of each blade's natural frequency from the mean value is then taken as a measure of the mistuning. However, the mistuning measured through this method may be significantly different from the mistuning present once the blades are mounted on the disk. This variation in mistuning can arise because each blade's frequency is dependent on the contact conditions at the attachment. Not only may the blade-broach contact differ from the blade-disk contact, but the contact conditions can also vary from slot-to-slot around the wheel or disk. Therefore, to accurately measure mistuning, it is desirable to develop methods that can infer the mistuning from the vibratory response of the blade-disk assembly as a whole. In addition, many blade-disk structural systems are now manufactured as a single piece in which the blades cannot be physically separated from the disk. In the gas turbine industry they are referred to as blisks (for bladed disks) or IBRs (for integrally bladed rotors). Thus, in the case of IBRs, the conventional testing methods of separately measuring individual blade frequencies cannot be applied and, therefore, it is desirable to develop methods that can infer the properties of the individual blades from the behavior of the blade-disk assembly as a whole.
Therefore, to accurately measure mistuning, it is desirable to develop methods or reduced order models that apply to individual blades or the blade-disk assembly as a whole. It is further desirable to use the mistuning values obtained from the newly-devised reduced order models to verify finite element models of the system and also to monitor the frequencies of individual blades to determine if they have changed because of cracking, erosion or other structural changes. It is also desirable that the obtained mistuning values can be analytically adjusted and used with the reduced order model to predict the vibratory response of the structure (or bladed disk) when it is in use, e.g., when it is rotating in a gas turbine engine, an industrial turbine, a fan, or any other rotating apparatus.